Subgroups with a small sum of element orders

A finite group $G$ is said to be a ${\mathcal{B}}_{\psi }$-group if $\psi (H)<\mathrm{\mid }G\mathrm{\mid }$ for any proper subgroup $H$ of $G$, where $\psi (H)$ denotes the sum of element orders of $H$. In this paper, we characterize the ${\mathcal{B}}_{\psi }$-groups up to the class of finite nilpotent groups. We also study the above inequality in the case of finite $P$-groups and Schmidt groups, respectively.